Sieve methods in group theory \Rmnum{3}: $\aut(F_n)$

Abstract

Let $\pi:\aut(F_n)\rightarrow \aut(\Z^n)$ be the epimorphism induced by the isomorphism $\Z^n \cong F_n/F_n'$ and define $\mathcal{T}_n:=\ker\pi$. We prove that the subset of $\mathcal{T}_n$ consists of all non-iwip and all non-hyperbolic elements is exponentially small.